The ρ parameter at three loops and elliptic integrals

Abstract

We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the ρ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.

Three-loop contributions to the ρ parameter and iterated integrals of modular forms

“…This distinguishes our computation from the results of refs. [45,46], where a novel class of special functions was introduced for the same class of integrals.…”

“…Before we discuss the computation of f and f 10 in the next section, let us mention that the system of differential equations satisfied by f (2) 8 and f (2) 9 was also analysed in refs. [45,46]. There the corresponding second-order differential operator was not directly in a form which matches the one for the sunrise graph in eq.…”

“…As a consequence, the homogeneous solutions of refs. [45,46] (i.e., the equivalent of our f (2) 8,h and f (2) 9,h ) take a different form, which cannot easily be recognised as being related to the sunrise graph.…”

“…This is at variance with the analytic results for Topology A of refs. [45,46], where new classes of functions were introduced.…”

“…First steps in this direction were taken in refs. [45,46], where the elliptic Feynman integrals of ref. [44] were computed in terms of a class of new transcendental functions, called iterative non-iterative integrals.…”